Global Universality of Singular Values in Products of Many Large Random Matrices
Boris Hanin, Tianze Jiang
TL;DR
This work establishes a global universality result for the singular-value distribution of products of independent random matrices in the simultaneous growth regime $n,N\to\infty$. By combining novel small-ball estimates for singular vectors with martingale concentration and a Doob decomposition, the authors prove non-asymptotic bounds that force the empirical distribution of the rescaled squared singular values to converge to the uniform law on $[0,1]$, under minimal moment and density conditions on the entry distribution $\mu$. A key feature is that universality emerges for the Lyapunov-exponent distribution once the matrix size grows with $N$, even though fixed-$n$ universality fails, highlighting a distinct global- versus local-statistics separation. The techniques yield quantitative control via $d_{KS}$ bounds and open pathways to extensions to complex matrices, with implications for understanding Jacobians in randomized systems and neural networks under broad randomness assumptions.
Abstract
We study the singular values (and Lyapunov exponents) for products of $N$ independent $n\times n$ random matrices with i.i.d. entries. Such matrix products have been extensively analyzed using free probability, which applies when $n\to \infty$ at fixed $N$, and the multiplicative ergodic theorem, which holds when $N\to \infty$ while $n$ remains fixed. The regime when $N,n\to \infty$ simultaneously is considerably less well understood, and our work is the first to prove universality for the global distribution of singular values in this setting. Our main result gives non-asymptotic upper bounds on the Kolmogorov-Smirnoff distance between the empirical measure of (normalized) squared singular values and the uniform measure on $[0, 1]$ that go to zero when $n, N\to \infty$ at any relative rate. We assume only that the distribution of matrix entries has zero mean, unit variance, bounded fourth moment, and a bounded density. Our proofs rely on two key ingredients. The first is a novel small-ball estimate on singular vectors of random matrices from which we deduce a non-asymptotic variant of the multiplicative ergodic theorem that holds for growing matrix size $n$. The second is a martingale concentration argument, which shows that while Lyapunov exponents at large $N$ are not universal at fixed matrix size, their empirical distribution becomes universal as soon as the matrix size grows with $N$.